Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

Shiping Cao, Hua Qiu

Abstract

We prove the uniqueness of self-similar $D_4$-symmetric resistance forms on unconstrained Sierpinski carpets ($\mathcal{USC}$'s). Moreover, on a sequence of $\mathcal{USC}$'s $K_n, n\geq 1$ converging in Hausdorff metric, we show that the associated diffusion processes converge in distribution if and only if the geodesic metrics on $K_n, n\geq 1$ are equicontinuous with respect to the Euclidean metric.

Uniqueness and convergence of resistance forms on unconstrained Sierpinski carpets

Abstract

We prove the uniqueness of self-similar -symmetric resistance forms on unconstrained Sierpinski carpets ('s). Moreover, on a sequence of 's converging in Hausdorff metric, we show that the associated diffusion processes converge in distribution if and only if the geodesic metrics on are equicontinuous with respect to the Euclidean metric.
Paper Structure (15 sections, 36 theorems, 145 equations, 4 figures)

This paper contains 15 sections, 36 theorems, 145 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The geometry of unconstrained Sierpinski carpets
  3. Review of resistance forms
  4. Convergence of resistance forms
  5. Weak convergence of processes
  6. A trace theorem
  7. Building bricks and boundary graph
  8. An upper bound estimate of the resistance metric
  9. A restriction result
  10. Proof of Theorem \ref{['thm43']}
  11. An estimate of effective resistances
  12. Uniqueness
  13. Weak convergence: sliding the $\mathcal{USC}$
  14. Basic geometric properties
  15. Proof of Theorem \ref{['thm71']}

Key Result

Theorem 1

Let $K$ be a $\mathcal{USC}$. There is a unique self-similar $D_4$-symmetric resistance form $(\mathcal{E},\mathcal{F})$ on $K$ such that the associated resistance metric $R$ is jointly continuous on $K\times K$ and satisfies $R(L_2,L_4)=1$.

Figures (4)

  • Figure 1: Unconstrained Sierpinski carpets ($\mathcal{USC}$'s).
  • Figure 2: More $\mathcal{USC}$'s.
  • Figure 3: The infinite graph $G_K$.
  • Figure 4: The $\mathcal{USC}$$K(z)$.

Theorems & Definitions (71)

  • Theorem 1
  • Theorem 2
  • Definition 2.1: Unconstrained Sierpinski carpets
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Definition 3.1
  • Lemma 3.2
  • Proposition 3.3
  • ...and 61 more